Geometrically irreducible $p$-adic local systems are de Rham up to a twist

TL;DR

The paper proves that geometrically irreducible p-adic local systems on smooth varieties over p-adic fields are de Rham after a Galois twist, linking local systems to de Rham representations and confirming cases of the Fontaine-Mazur conjecture.

Contribution

It establishes that such local systems become de Rham after a Galois twist and connects their properties to the Fontaine-Mazur conjecture, using p-adic Simpson and Riemann-Hilbert correspondences.

Findings

Any geometrically irreducible local system becomes de Rham after a Galois twist.

A local system is Hodge-Tate if its stalk at one point is Hodge-Tate.

Galois action on the pro-algebraic fundamental group is de Rham.

Abstract

We prove that any geometrically irreducible $\overline{\mathbb{Q}}_p$-local system on a smooth algebraic variety over a $p$-adic field $K$ becomes de Rham after a twist by a character of the Galois group of $K$. In particular, for any geometrically irreducible $\overline{\mathbb{Q}}_p$-local system on a smooth variety over a number field the associated projective representation of the fundamental group automatically satisfies the assumptions of the relative Fontaine-Mazur conjecture. The proof uses $p$-adic Simpson and Riemann-Hilbert correspondences of Diao-Lan-Liu-Zhu and the Sen operator on the decompletions of those developed by Shimizu. Along the way, we observe that a $p$-adic local system on a smooth geometrically connected algebraic variety over $K$ is Hodge-Tate if its stalk at one closed point is a Hodge-Tate Galois representation. Moreover, we prove a version of the main theorem for local systems with arbitrary geometric monodromy, which allows us to conclude that the Galois action on the pro-algebraic completion of the fundamental group is de Rham.